A268216 Triangle read by rows: T(n,k) (n>=2, k=3..n+1) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,1,2,...,s,n} where s is the size of the largest proper open set in t.

2, 3, 6, 4, 12, 24, 5, 20, 60, 120, 6, 30, 120, 360, 720, 7, 42, 240, 840, 2520, 5040, 8, 56, 336, 1680, 6720, 20160, 40320, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800

Offset

2,1

Links

Table of n, a(n) for n=2..46.

G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.

Example

Triangle begins:
   2;
   3,  6;
   4, 12,  24;
   5, 20,  60,  120;
   6, 30, 120,  360,   720;
   7, 42, 240,  840,  2520,   5040;
   8, 56, 336, 1680,  6720,  20160,  40320;
   9, 72, 504, 3024, 15120,  60480, 181440,  362880;
  10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800;
  ...

Mathematica

i = 1; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0, n - i - 1}], {n, 2, 9}] // Grid (* Geoffrey Critzer, Feb 19 2017 *)

Crossrefs

Row sums give A038156. A008279 (main entry).

Triangles in this series: A268216, A268217, A268221, A268222, A268223.

Cf. A282507.

Sequence in context: A097275 A130879 A119741 * A346928 A375761 A245712

Adjacent sequences: A268213 A268214 A268215 * A268217 A268218 A268219

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 29 2016

Extensions

Definition clarified by Geoffrey Critzer, Feb 19 2017

Status

approved